Planetary Temperature, the Cloud Blanket Effect
and the Greenhouse Effect for Venus

A planet's surface temperature is established by a balance of the incoming
radiation which is absorbed and the outgoing radiation which escapes into space. Some radiation from the surface is captured by greenhouse
gases in the atmosphere and they radiate in all directions, some of which
reaches the planet's surface. But this is not the only mechanism by which
energy radiated from the surface gets returned to the surface. Radiation
can be reflected from the underside of clouds. This cloud blanket effect
might compete with the greenhouse effect in raising a planet's surface
temperature.

It is a common experience to find that a night with cloud cover is warmer
than a night with clear skies, all other things being equal. The greenhouse
effect would be the same on both nights so it is the reflectance of surface radiation off the undersides of clouds that produces the warming.

The planet Venus is surrounded completely by clouds of sulfur dioxide and droplets of sulfuric acid. These
clouds reflect back about 75 percent of the Sun's radiation that falls upon
them. The atmosphere of Venus is about 96 percent carbon dioxide and is
quite dense so there is a strong greenhouse effect, but there might also be a
strong cloud blanket effect. It would be of interest to determine how much of the
elevated temperature of Venus, about 470°C, is due to the greenhouse effect
and how much due to the cloud blanket effect.

The Energy Balance Model

A planet's surface absorbs some of the short wavelength (visible and ultraviolet) radiation impinging
upon it and reflects some back out into space. The surface radiates long wavelength (infrared) radiation out into space based on its temperature.

Let α be the proportion of short wavelength radiation reflected from
a planet's surface. This is its albedo. Let σ be the solar constant,
the intensity of the incoming solar radiation be unit area perpendicular to the
radiation. The intensity of the outgoing
long wavelength radiation is given kT4, where T is the absolute
temperature of the surface in degrees Kelvin and k is a constant, known as
the Stefan-Boltzmann constant and has a value of 5.67×10-8 Wm-2K-4. If R is the radius of the planet the net energy absorbed
at the surface is σ(1−α)(πR²). Assuming the planet's
surface has an average temperature T, the outgoing energy
is kT4(4πR²). For balance then

4kT4(πR²) = σ(1−α)(πR²)
which upon division of both sides by the surface area 4πR² gives
kT4 = (σ/4)(1−α)
and hence
T = (σ(1−α)/(4k))1/4

The actual area of the planet is thus irrelevant and hereafter all computations
will be per square meter of surface area.

Cloud Cover

Let β be the albedo of a cloud cover. Then a share of β of the
incoming radiation is reflected and a share of (1-β) passes through the
clouds. For an atmosphere transparent to the incoming radiation the energy
incident upon the surface is σ(1-β). A share (1-α) is absorbed
and σ(1-β)α is reflected away from the surface. This impinges
upon the underside of the clouds and a share of β is reflected back down to
the surface. The energy of this reflectance is σ(1-β)αβ.
The next round of reflectances results in σ(1-β)(αβ)²
arriving at the surface. At the next round it is σ(1-β)(αβ)³.

The surface absorbs a share (1-α) of this and thus the total energy
absorbed by the surface from the short wavelength radiation is

σ(1-α)(1-β)/(1−αβ)

On the other hand, a unit of surface radiates 4kT4 up to the underside of the clouds. Assuming the same reflectances for long wavelength radiation as
for short wavelength, radiant of energy 4kT4β comes back. A round
of reflectances from the surface to the clouds and back again results in
(4kT4β)(αβ) at the surface and likewise for subsequent
rounds of reflectances. The total returning radiation is then

(4kT4β)(1 + αβ+(αβ)²+…)
which evaluates to
(4kT4β)/(1−αβ)

Of this the surface absorbs a share (1-α). Thus the total energy
absorbed at the surface is

σ(1-α)(1-β)/(1−αβ) +
(4kT4β(1-α))/(1−αβ)

For equilibrium this has to equal the energy radiated away from the surface; i.e., 4kT4. Thus energy balance at the surface then requires

This is simply the equilibrium temperature which would prevail with no cloud
cover. This result is predicated on the assumption that the cloud and surface
albedoes for infrared radiation are the same as for visible and ultraviolet
radiation.

The
cloud cover does not induce a higher temperature.

A Numerical Illustration

The albedo of Mars is about 0.25. Without clouds, vegetation and water that
would be the albedo of Earth and also of Venus. The albedo of Venus is about 0.75
and this is an appropriate value for the cloud albedo β.

The solar constant for Venus is 2611 watts per square meter of planetary cross
section area.

If Venus had no clouds and no greenhouse effect from its atmosphere its
temperature would be

T = ((2611/4)*(0.75) /(5.67×10-8)1/4305°K = 32°C

If there are no clouds 653 watts of energy would blaze down on each square meter of surface. 163 watts would be reflected out into space and 490 watts would be absorbed. To radiate 490 watts out into space the equilibrium temperature would
have to be 305°K.

Now consider the case of Venus with a 0.75 albedo cloud cover but no
greenhouse gases in the atmosphere.
With 2611/4=653 watts per square meter of cloud surface area coming in, 490 watts is reflected back out
into space and 163 watts passes through the cloud and down to the surface.
The surface reflects 25 percent, or 41 watts, and absorbs 122 watts. But of the
41 watts reflected from the surface 31 watts are reflected back from the
underside of the clouds. At the surface 7.65 watts are re-reflected and 23 watts absorbed. So the short wavelength energy impinging upon the surface is 163 watts and the
energy leaving by reflectance is 41 watts. The total energy impinging upon the
surface is 163/(1-0.75*0.25)=163/(1-0.1875)=163/(0.8125)=201 watts and the amount
absorbed is 150 watts.

At a temperature of 305°K the surface is radiating energy equal to
(5.67×10-8)(86.5×108=491 watts per square meter.
Of these 491 watts, 368 watts are reflected back. This 368 watts gets reflected
and re-reflected until the total energy impinging upon the surface is 368/0.8125=453 watts. Of these 453 watts, 340 watts are reabsorbed by the surface. The total
energy absorbed per square meter would be 150+340=490 watts. (This is slightly different from the 491 watts radiated from a square meter of surface because of independent rounding.) So the surface temperature of Venus with no greenhouse effect would be the same with and without the clouds. However note that with the clouds there is
a cloud blanket effect but it only offsets the effect of the reflectance of
solar radiation from the outside of the clouds. The visible light on Venus is
quite dim but the intensity of the infrared radiation quite large.

The actual average temperature of Venus is
about 740°K or 470°C, so the Venus is about 435° warmer due to its
greenhouse effect and other factors.
According to Barton Paul Levenson on
his webpage How to Estimate Planetary Temperatures at
bartonpaullevenson.com/NewPlanetTemps.html,
the greenhouse effect of Venus' atmosphere accounts for a temperature of
631°K. That is 326°C above what it would be with no clouds and no
greenhouse effect. What makes the greenhouse so much more powerful than the
cloud blanket effect is that the greenhouse gases are transparent to the
short wavelength radiation whereas a cloud reflects out short wavelength
radiation as well as reflecting in the long wavelength radiation.

Conclusion

If the albedoes for clouds and surface are the same for long wavelength (infrared) radiation as for short wavelength radiation (visible and ultraviolet) then the cloud cover of a planet does not raise its equilibrium temperature.
The effect of the cloud cover to retain more of the long wavelength radiation
is exactly offset by the clouds shielding the planet's surface from
the solar short wavelength radiation.